Math Problem of the Day

Arjun has been playing video games until very late in the night, sometimes forgetting to post his MPOTD problem. When playing those games, he decides to host a server so his friends can connect faster, but he needs to find the optimal placement to minimize the overall ping from all his friends. Conveniently, everyone’s houses are arranged on a grid. TJHSST is at (10, 10), Paul is at (8, 5), Minh is at (4, 9), Arjun is at (2, 10), Fareez is at (7, 2), and Albert is at (1, 3). There are a string of empty warehouses on the function $$ f(x) = \frac{x^4 + 3x^3 + x + 3}{100} $$ along which the server can be placed. What is the minimum possible sum of the ping times between each house and the server, if ping is defined as the round trip time and packets move at a constant rate of $\frac{1}{5} \frac{units}{ms}$?